From 82e6cf7d1046d6cee16f7e8b044ec33e7ec6c4b7 Mon Sep 17 00:00:00 2001
From: Nguyễn Gia Phong <vn.mcsinyx@gmail.com>
Date: Sun, 16 Feb 2020 14:26:55 +0700
Subject: [usth] Numerical Method is MATH2.4

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-\documentclass[a4paper,12pt]{article}
-\usepackage[english,vietnamese]{babel}
-\usepackage{amsmath}
-\usepackage{lmodern}
-\usepackage{hyperref}
-\usepackage{tikz}
-
-\newcommand{\exercise}[1]{\noindent\textbf{#1.}}
-\renewcommand{\thesection}{\Roman{section}}
-\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
-
-\title{Numerical Method: Labwork 2 Report}
-\author{Nguyễn Gia Phong--BI9-184}
-\date{Fall 2019}
-
-\begin{document}
-\maketitle
-\setcounter{section}{2}
-\section{Polynomial}
-\exercise{1.c} At the time of writing, function \verb|fzero|
-in Octave have not support the \verb|Display| option
-just yet\footnote{Bug report: \url{https://savannah.gnu.org/bugs/?56954}}.
-However, the implementation of this option is rather trivial,
-thus I made a quick patch (which is also attached at the bug report).
-Using this, one can easily display all the iterations as followed:
-
-\begin{verbatim}
-octave:1> fzero (@(x) x.^2 - 9, 0, optimset ('display', 'iter'))
-
-Search for an interval around 0 containing a sign change:
-Func-eval 1,  how = initial,  a = 0,  f(a) = -9,  b = 0,  f(b) = -9
-Func-eval 2,  how = search,  a = 0,  f(a) = -9,  b = 0.099,  f(b) = -8.9902
-Func-eval 3,  how = search,  a = 0,  f(a) = -9,  b = 0.1025,  f(b) = -8.98949
-Func-eval 4,  how = search,  a = 0,  f(a) = -9,  b = 0.095,  f(b) = -8.99098
-Func-eval 5,  how = search,  a = 0,  f(a) = -9,  b = 0.11,  f(b) = -8.9879
-Func-eval 6,  how = search,  a = 0,  f(a) = -9,  b = 0.075,  f(b) = -8.99437
-Func-eval 7,  how = search,  a = 0,  f(a) = -9,  b = 0.15,  f(b) = -8.9775
-Func-eval 8,  how = search,  a = 0,  f(a) = -9,  b = 0,  f(b) = -9
-Func-eval 9,  how = search,  a = 0,  f(a) = -9,  b = 0.35,  f(b) = -8.8775
-Func-eval 10,  how = search,  a = 0,  f(a) = -9,  b = -0.4,  f(b) = -8.84
-Func-eval 11,  how = search,  a = 0,  f(a) = -9,  b = 1.1,  f(b) = -7.79
-Func-eval 12,  how = search,  a = 0,  f(a) = -9,  b = -4.9,  f(b) = 15.01
-
-Search for a a zero in the interval [-4.9, 0]:
-Func-eval 13,  how = initial,  x = 0,  f(x) = -9
-Func-eval 14,  how = interpolation,  x = -1.83673,  f(x) = -5.62641  (NaN%)
-Func-eval 15,  how = interpolation,  x = -3.36837,  f(x) = 2.3459  (141.7%)
-Func-eval 16,  how = interpolation,  x = -3.19097,  f(x) = 1.1823  (-49.6%)
-Func-eval 17,  how = interpolation,  x = -2.99725,  f(x) = -0.0164972  (-101.4%)
-Func-eval 18,  how = interpolation,  x = -3.00258,  f(x) = 0.0154927  (193.9%)
-Func-eval 19,  how = interpolation,  x = -3,  f(x) = 3.07975e-07  (-100.0%)
-Func-eval 20,  how = interpolation,  x = -3,  f(x) = -7.10543e-15  (-100.0%)
-Func-eval 21,  how = interpolation,  x = -3,  f(x) = 5.32907e-15  (169.7%)
-
-Algorithm converged
-
-ans = -3.0000
-\end{verbatim}
-
-To answer the question in part b, (since I believe these parts are linked
-to each other), the current implementation of \verb|fzero| search for
-the second bracket over quantitative chages below if \verb|X0| if it is a
-single scalar, thus $[-4.9, 0]$ is gotten and the found solution is negative:
-
-\begin{verbatim}
-[-.01 +.025 -.05 +.10 -.25 +.50 -1 +2.5 -5 +10 -50 +100 -500 +1000]
-\end{verbatim}
-
-\section{Non-linear Systems}
-\exercise{1.a} These statements were used to plot the given functions:
-\begin{verbatim}
-ezplot(@(x1, x2) x1 .^ 2 + x1 .* x2 - 10)
-hold on
-ezplot(@(x1, x2) x2 + 3 .* x1 .* x2 .^ 2 - 57)
-\end{verbatim}
-
-As shown in the graphs (where $x_1^2 + x_1 x_2 = 10$ are the blue lines
-and $x_2 + 3 x_1 x_2 = 57$ are the yellow ones), the solutions of $(x_1, x_2)$
-are quite close to $(2, 3)$ and $(4.5, -2)$.
-
-\begin{figure}[!h]
-  \centering
-  \scalebox{0.37}{\input{2a.tikz}}
-\end{figure}
-
-I would also like to note that I am personally impressed how gnuplot
-(which is utilised by Octave) is able to export to TikZ graphics with ease.
-\end{document}
-- 
cgit 1.4.1

